Car accident, need help!

I had a car accident last Saturday and the worst thing is I bought this new car for just a month (financing with a bank). The good thing is none of my families is injured though it was a very dangerous accident that occured on the Interstate 10. And I was driving too fast and was really very very tired at that moment.

I want to ask the friends here for a couple of questions about car insurance.
First of all, the insurance company sent us the initial estimation paper which said it was very probable that my car might be a total loss. They want us to sign an release of our "power of attorney". Thus, they could evaluate the market/cash value of my car. I called the repair shop and they don't think my car was that severe
damaged. I want to ask friends here I should sign the paper or not?

Second, if my car were at a total loss will the insurance company compensate me a brand new car just like mine? I couldn't figure out how that will work. It is a brand new 2003 Japanese car that costs about $20,000.

Third, if my car were at a total loss how will my insurance premium change in the future years?

I felt really exhusted and scared in the past few days and I do need some help from the friends here. Many many thanks.